Linear Inequations, Replacement Set and Solution Set

Linear Inequations, Replacement Set and Solution Set

Linear Inequations

We know that in a statement in which two sides are equal is known as an equation. An equation whose highest degree is one is known as linear equation.
For example, 2x + 3 = 5 and  5a – 7 = 0, etc.
A statement in which two sides are related by the symbols >, <, or are called inequation. An inequation whose highest degree is one is known as linear inequation.
For example, 3x – 2 < 12, 4y + 7 15, etc.
In general, a linear inequation can be written as
1. ax + b < 0
2. ax + b > 0
3. ax + b 0
4. ax + b 0
where a, b are real numbers and a 0.

Replacement Set and Solution Set

The set from which the values of the variable in a linear inequation are chosen is called the replacement set or universal set.
The set of elements of the replacement set which satisfy the inequation when substituted for the variable is called the solution set or truth set. The solution set depends upon the replacement set.
For the inequation x 5,
1.      If replacement set is {3, 4, 5, 6, 7, 8, 9}, then the solution set is {5, 6, 7, 8, 9}.
2.      If replacement set is {all natural numbers}, then the solution set is {all natural numbers greater than or equal to 5}
3.      If replacement set is {–2, –1, 0, 1, 2, 3, 4}, then the solution set is ϕ.

Properties of Inequations

1.     Addition or subtraction of the same integer from both sides of an inequation does not change the inequation, i.e., if a > b, then a + c > b + c and a – c > b – c.
2.      Multiplying or dividing by the same positive integer on both sides of an inequation does not change the inequation, i.e., if a > b and c is any positive integer, then a × c > b × c and a/c > b/c.
3.     Multiplying or dividing by the same negative integer on both sides of an inequation reverses the sign of inequality, i.e., if a > b and c < 0, then a × c < b × c and a/c < b/c.

4.      If the sides of an inequation are interchanged, then the sign of inequality also reverses, i.e., if a > b, then b < a.

Example: Solve 5x + 2 22, x ϵ W.
Solution: Given inequation is, 5x + 2 22
5x + 2 – 2 22 – 2       (Subtracting 2 from both sides)
5x 20
x 4        (Dividing both sides by 5)
Thus, the solution set for 5x + 2 22, x ϵ W is {0, 1, 2, 3, 4}.